The class blog for Math 3010, fall 2014, at the University of Utah

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Diophantus of Alexandria is thought of as “the father of algebra,” a subject that many students in high school have to endure. Though he had an impact and is part of mathematical history, very little is known about Diophantus. “The father of algebra” lived in Alexandria, Egypt, possibly around 200 A.D. to around 290 A.D. One of Diophantus’ contributions in mathematics was his work Arithmetica, which included 13 books but many have been lost and only 6 out of the 13 survived. He also worked on The Porisms. Like much of his other work, this book was also lost. Much of his work impacted mathematics. Specifically his work on Arithmetica led to “employ algebra as a modern style.” He was the first to use symbols for unknowns in arithmetic operations.

Diophantus’ Arithmetica helped many mathematicians to construct their own ideas. The 6 books from Arithmetica that have been found each contain problems with their solutions. Most of the problems in Arithmetica are algebraic but Diophantus was not the first one to work with algebraic problems. What is interesting about the problems and solutions in Arithmetica is that Diophantus never used one general idea for solving his problems. He had a different way of solving each problem. The book also contains problems that have determinate (a unique solution) and indeterminate solutions (more than one solution). When looking through Arithmetica many of the solutions are fairly complicated and the steps in the solutions are puzzling. It can be seen that in Diophantus was able to improve some notation in algebra, but how the algebra was solved still needed some improvement.

Diophantus included in his book about 130 algebraic problems. More specifically, any solutions that contained negative or irrational square roots were considered useless to Diophantus because he did not believe in negative numbers. An example of this (that we would see in any high school text book) could be shown with the equation, 4 = 6x + 40. This was considered as an absurd equation because it would lead to a meaningless answer. The solution to this would be -6 and Diophantus did not consider negative numbers as things that were real. When we learn and study about quadratic equations, we know that the result could possibly have two solutions. “There is no evidence to suggest that Diophantus realized that a quadratic equation could have two solutions. However, the fact that he was always satisfied with a rational solution and did not require a whole number is more sophisticated than we might realize today” write J J O’Connor and E F Robertson in Diophantus in Alexandria.

Problems that Diophantus worked with included Pythagorean triples. Image: Gustavb, via Wikimedia Commons.

Diophantus worked on equations including Pythagorean triples. These are triples of numbers, like (3,4,5), for which x2+y2=z2. Image: Gustavb, via Wikimedia Commons.Another type of algebraic problem he would have worked with would have involved two integers such that the sum of their squares is a square. This is what we know today as a Pythagorean triple. An example of this would be x2 + y2 = z2, where x = 3 and y = 4 giving z = 5. Another example would be x = 5 and y =12 giving z = 13. He also worked with two integers such that the sum of their cubes is a square. This would look like x3 + y3 = z2. An example of this would be x = 1 and y = 2, giving z = 3. Diophantus tried to determine if such problems, like these or similar to these, had any solutions or none at all.

Arithmetica contributed to the development of some math ideas, for example, Fermat’s Last theorem. Pierre De Fermat claimed that a generalization of the equation in Arithmetica had no solutions. In the margin of the book he wrote that he “had found a marvelous proof to a proposition, which however the margin is not large enough to contain.” Of course there was no discovery of any proof from Fermat. Though Arithmetica was written in the 3rd century A.D., it impacted the mathematics world most notably in the 90’s when Andrew Wiles proved Fermat’s Last theorem. What can be seen from Diophantus’ work is that math can unfold over time and continue with development.

Though many of Diophantus’ ideas contributed to further ideas in math, some might say that he is not the father of algebra. Many of the methods he used for solving linear and quadratic equations go back to Babylonian and Egyptian mathematics. For this reason, mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the Father of Algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.” Either way, Diophantus impacted the math world with his book Arithmetica.

The modern city of Kaliningrad, Russia once had 7 bridges spanning 4 islands. Back when it still had its famous 7 bridges, the city was called Königsberg. Because of the unique configuration of the bridges, the question was posed: Is it possible to travel across every bridge exactly once? The answer was no. This problem was studied by Euler, and in honor of his solution is called the Euler path problem. He broke down the problem to a graph, or a system of nodes connected by edges. He proved that in order for a path that traverses every edge once to exist, only 0 or 2 nodes can have an odd number of edges leading to it. His solution is elegant and makes solving problems such as the Königsberg bridge problem very easy.

William Rowan Hamilton, however, struggled with a more difficult problem. Rather than using every edge exactly once, he posed the problem of visiting every node exactly once. While this is easy for small graphs, it is very difficult to determine if an arbitrary graph (of, say, 1000s of nodes) has a route that visits every node exactly once. At this point, with over 1000 nodes, we have exited the realm of human feasibility. We could make the assumption that it would be easy to solve this with a computer, but that assumption would be false. As it turns out, there is no known algorithm that efficiently solves this problem.

To make matters worse, the problem can be checked really easily. It is almost trivial for a computer to determine that a Hamiltonian path is valid. This brings us to an infuriating question: Because a problem can be checked easily, shouldn’t it be just as easy to solve? Unfortunately, the answer is still unknown. This is called P vs. NP and is one of the most important questions of the entire field of Computer Science. In fact, it is such an important question that the Clay Institute of Mathematics selected it as one of their 7 Millennial Prize Problems. Only 1 of these problems has been solved so far, despite each solution coming with a prize of $1,000,000. While it is generally believed that P =/ NP, it has not been conclusively proven impossible.

The Hamiltonian path problem is part of a set of problems called NP-Complete. This means that while it is easy to check the solution of any of these problems, it is nigh-impossible to come up with a solution. Other problems in this category are equally famous and include the Knapsack problem, the Traveling Salesman problem (which is basically the Hamiltonian path problem made more difficult), and the Boolean Satisfiability problem. Other, more obscure problems also belong category, such as: “Does an arbitrary Minesweeper puzzle have a solution?” or “Is it possible to travel between two given points in a randomly generated Pokemon map?” or “Does an arbitrary Legend of Zelda block-pushing puzzle have a solution?” Each of these problems can be reduced to an existing NP-Complete problem, and to make matters worse, every NP-Complete problem can be reduced to any other NP-Complete problem.

If a positive solution to the P vs NP problem was found (so that P=NP), the implications would be staggering. First, every NP-complete problem would be solvable in polynomial time. Difficult algorithms, from complicated graphics meshing to image processing, would be made easy. Almost every cryptographic system would be breakable quickly, including AES, the gold standard of symmetric cryptography. This would have implications outside of the realm of computer science – for example, RNA protein folding is also NP-Complete. With a new, super-efficient algorithm, the flu, hepatitis C, and every other RNA-based disease could be completely eradicated. Huge strides could be made in the field of genetics, particularly in the manufacture of human proteins and enzymes.

If a negative solution to the P vs NP problem was found, the upside is that someone gets a million dollars. That’s probably the best thing that will happen. Suddenly, we will no longer have the hope of efficient algorithms. We can settle for less-than-perfect algorithms and focus on more productive areas of research. At the very least, the field of computer science will be able to say that they have answered one of the most difficult problems the entire area of study has ever dealt with.

In class we discussed the famous mathematician, Sophie Germain. Upon discussing her I had thoughts that I’m sure many women in math have also had, “why have I never heard of a female mathematician until now?” This caused me to look into female mathematicians throughout history and what their accomplishments were.

Image: Jules Maurice Gaspard, via Wikimedia commons.

The first known woman involved in mathematics is Hypatia. She was from Alexandria, Egypt, and was born sometime between 351-355 AD and died in 415. Her father, Theon, played a major role in her becoming the mathematician that she was. Theon was a professor at the University of Alexandria and was determined to have a “perfect child”[1]. This led to him teaching Hypatia to be very well rounded in all subjects, including math. Hypatia went on to teach at the University of Alexandria and became very popular. Her lectures often were on Diophantus’ “Arithmetica” and the techniques he used. One of the more interesting things about Hypatia was that she was not afraid to go to a group of men, and that she was highly admired by those men [2].

Elena Cornaro Piscopia was not only a mathematician, but was highly skilled in music, philosophy, and language. She was from Venice, Italy, and lived from 1646-1684. She was the first woman to ever receive a doctorate degree, in philosophy, not math, and was only thirty-two years old [3]. Another female mathematician who lived near the same time period of Piscopia was Émilie du Châtelet. She was a French mathematician and physicist who lived from 1706-1749. Her most famous work was the translation of Newton’s Principia Mathematica [4].

Sophie Germain is probably one of the most well-known female mathematicians. She was born in Paris, France 1776. She started her journey into mathematics around the age of 13 when she came across mathematical texts in her father’s library. She began to study them relentlessly, even though her parents did not support her. She began to have an interest in number theory after the release of Legendre’s Essai sur la théorie des nombres. From this point on she began to make huge progress in the field of number theory. As we all know she helped make advancements in the proof of Fermat’s Last Theorem, but she also went on to win an Academy Prize for her work in elasticity. While all this went on, I still find it interesting how she had support from many famous male mathematicians, such as Legendre, Gauss, and Poisson, yet when it came to publishing their own work or helping her further her own they were not as supportive. Germain was never mentioned in Poisson’s work, yet she helped him often and he had access to all of her work. Legendre began to help Germain in her work that won her the Academy Prize, but as time went on he refused to help her anymore. Women were not accepted as scholars during this time, so much so that Germain couldn’t even attend the Academy Prize sessions because she was a woman and not the wife of a male mathematician. It is truly amazing that through all the suppression she experienced that she was able to overcome it all to become one of the most well known female mathematicians ever [5].

Another female mathematician from a more current time period was Emmy Noether. She was born in Germany in 1882 and died in America in 1935. Noether was an algebraist and is known for her work in topology. She worked on Algebraic Invariant Theory and allowed the study of the relationships among the invariants to be possible. Invariant Theory deals with action of groups on algebraic varieties from the point of view of their effect on functions [7]. She made huge progress on ascending and descending chain conditions. A partially ordered set satisfies the ascending chain condition if every strictly ascending sequence of elements eventually terminates and it satisfies the descending chain condition if every strictly decreasing sequence of elements eventually terminates [9]. Most objects in abstract algebra that satisfy these conditions are called Noetherian after her. Some of these “Noetherians” include Noetherian induction, Noetherian modules, and Noetherian rings. Noether also did work in physics and published Noether’s First Theorem, which states that every differentiable symmetry of the action of a physical system has a corresponding conservative law, which she also proved [8]. She not only solved the problem for general relativity, but also determined the conserved quantities for every system of physical laws that possesses some continuous symmetry [6]. Noether’s advancements in math have lead others to call her the most important woman in the history of mathematics [6].

Lastly I will discuss a more current female mathematician, Maryam Mirzakhani. She was born in 1977 in Tehran, Iran. She has received many awards in her lifetime; she competed in two different International Mathematical Olympiads and received gold medals at both. She is also the first Iranian student to receive a perfect score at one of these Olympiads. In 2014 she was awarded the Fields Medal, an award that is given out every four years, to up to maybe four mathematicians under the age of forty. The award is often compared to the Nobel Prize of mathematics. She is the first female and Iranian to ever win the award [10]. Jordan Ellenberg explained her research when she won her award:

… [Her] work expertly blends dynamics with geometry. Among other things, she studies billiards. But now, in a move very characteristic of modern mathematics, it gets kind of meta: She considers not just one billiard table, but the universe of all possible billiard tables. And the kind of dynamics she studies doesn’t directly concern the motion of the billiards on the table, but instead a transformation of the billiard table itself, which is changing its shape in a rule-governed way; if you like, the table itself moves like a strange planet around the universe of all possible tables … This isn’t the kind of thing you do to win at pool, but it’s the kind of thing you do to win a Fields Medal. And it’s what you need to do in order to expose the dynamics at the heart of geometry; for there’s no question that they’re there [10].

Mirzakhani is a modern day mathematician we can all look up to; she has accomplished things that no mathematician has, not just female mathematicians.

Overall throughout looking at the history of some famous women in math I have realized how amazing these women truly were. They were able to overcome many obstacles and repression to further advance a subject that they love, and I love: math.

Nearly every country had its own special role in the development of mathematics. Many stem from one another, building off of past achievements to contribute to what we now would use in modern mathematics. There are few countries that were able to develop their own mathematical theories without being derived from past work. The Japanese in particular is one of the few that stands out, in such that it is distinguished from Western mathematics. During the 1870s Japanese mathematics was given the term “wasan”, which translates to “Japanese calculation”. This was the term that distinguished Japanese mathematics theory from Western mathematics (“yōsan”). The term was used after the Edo Period (1603 – 1867), when Japan was still isolated from the rest of the world. It wouldn’t be until the Meiji Era (1868 – 1912) when that isolation ended and Japan opened up to the West, leaving the ideas of wasan behind.

The first noted mathematician in Japanese history is Mori Kambei, the teacher of Japanese mathematics. (“Mori” is the family name, so he will be referred to by this.) As expected from one of the most prominent teachers in the country, Mori had started a school in Kyoto and also wrote several books that involved arithmetic and the use of an abacus. One of his well-known students had written the mathematical text Jinkōki, one of the oldest documents written on elementary mathematics for everyday use. This student was known as Yoshida Mitsuyoshi. (“Yoshida” is the family name, so he will be referred to by this.) Yoshida was an exceptional mathematician who published his work during the Edo Period. He and his fellow students Imamura Chishō and Takahara Kisshu were known as “The Three Arithmeticians”, primarily because they were Mori’s most prominent students. Yoshida’s Jinkōki dealt with soroban arithmetic (abacus arithmetic), including square and cube root operations.

Seki Takakazu, Image: uploaded by F. Lembrez to Wikimedia Commons.

Around the same time calculus was developed in Europe, Seki Takakazu founded what was known as “enri” (circle principles). (“Seki” is the family name, so he will be referred to by this.) These principles served the same purpose as Western calculus. This system was Japan’s foundation for the development of wasan. Seki was known as “Japan’s Newton”, who created a new algebraic notation system and worked on infinitesimal calculus and Diophantine equations. All of Seki’s work was independent, unlike his European counterparts (Gottfried Leibniz and Isaac Newton, just to name a few). However, much of his work paralleled European achievements; as an example, he was credited with the discovery of Bernoulli numbers (sequence of rational numbers that appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler-Maclaurin formula, and in expressions for certain values of the Riemann zeta function). Seki’s work were more or less based on or related to algebra with numerical methods, polynomial interpolation (and its applications), and indeterminate integer equations. He also worked on the development of general multi-variable algebraic equations and elimination theory– the equivalent of Gaussian elimination to solve linear equations. This timeline nearly reflects Western discovery of mathematical theories, just a few decades earlier.

To follow up on elimination theory, Seki developed the notion of determinant. Seki’s pupil, Takebe Katahiro, came up with the resultant and Laplace’s formula of determinant for the nxn case. Seki’s first manuscript treated only up to the 3×3 case. A large part of the problems treated at the time became solvable in principle, and the elimination method would flounder under a very large computational complexity. When the elimination is completed, the real roots of a single variable equation had to be found numerically. Diverging from elimination theory, Seki also studied the properties of algebraic equations. The most prominent were the conditions for the existence of multiple roots based on the discriminant (the resultant of a polynomial and its derivative, which was the order (h) term in f (x + h) accessible through the binomial theorem). Seki had also contributed to the calculation of pi, with an approximation that was correct to the 10th decimal place. This approximation was found using what is now known as the Aitken’s Delta-Squared Process, a series acceleration method used for accelerating the rate of convergence of a sequence.

Many of Mori’s works were succeeded by Yoshida, whose work was succeeded by Seki, whose work was succeeded by his own students and so on. As each generation continued to work on wasan, the integration of yōsan progressively established a foundation in Japanese mathematics. European ideas helped develop Japanese arithmetic, which continued to produce work nearly identical to older Western discoveries. Aside from Mori, Yoshida, and Seki, there were several other Japanese mathematicians who significantly contributed to wasan. If you’re interested, these individuals included Takebe Kenko, Matsunaga Ryohitsu, Kurushima Kinai, Arima Raido, Fujita Sadasuke, Ajima Naonobu, Aida Yasuaki, Sakabe Kōhan, Fujita Kagen, Wada Nei, Shiraishi Chochu, Koide Shuki, Omura Isshu, and many more.

Plimpton 322 is a historical math document displaying Babylonian mathematics on a clay tablet. I had never heard of such a tablet or seen an artifact similar to Plimpton 322. From watching the documentary in class on Plimpton 322, I had learned that it was a tablet with a list of Pythagorean triples. The tablet was discovered somewhere unknown in Iraq. Plimpton 322 is similar to math that we study today. It displays a numeric with a different base. I began to wonder what other historical mathematical artifacts had existed around the world and how they were used in their societies. Also what were the similarities and differences when comparing them to the Plimpton 322? Studying and learning about historical math artifacts can reveal our math heritage and explain why we learn and need math.

To start understanding how math was used thousands of years ago , we can go back to southern Africa in 35,000 B.C. One of the oldest mathematical artifacts that has been discovered in the world is the Lebombo bone. The bone was a baboon’s fibula bone with 29 markings that are believed to represent a lunar calendar. This bone was found in a cave in Lebombo, between South Africa and Swaziland. The bones were marked with important figures that recalled events yearly. Similar to the Plimpton 322, symbols that had a meaning were carved on an object to create a permanent marking. The reasons these artifacts were used, though, were completely different. These bones were used to keep track of yearly events. Of course, Plimpton 322, which is more than 30,000 years younger than the Lebombo bone, was used for a more advanced purpose. It tracked and listed Pythagorean triples by using a numbered system that the Babylonians had developed.

An example of what a quipu looked like, with knots and different colors. Image: A. Davey, via Flickr.

Though Plimpton 322 and the Lebombo bone used a mathematical system of markings on a surface to represent something significant to them, other cultures created artifacts that did not require anything to be written down. I came across the Quipu, which according to April Holloway is “the ancient mathematical device of the Inca.” The quipu is different from Plimpton 322 and Lebombo bone because it shows that mathematics doesn’t have to be written down to be significant. It was discovered in the valley of Canete close to Lunahuana, Peru. This device stands out because this Inca artifact was created out of various material that includes: llama or alpaca hair or cotton cords. These strands were colored, spun and plied. Maybe you are wondering, how did this artifact use math and how was it used in its society? April Holloway states that the strands “contained numeric and other values encoded by knots in a base ten positional system.” There was no limit on the number of cords each quipu could have had. Quipus ranged from having a few to 2,000 strands. The color, the amount knots, and the type of material used, all represented something specific to the Inca society because it was “both statistical and narrative information.” There have been 200 found but they are not older than 650 A.D. and because these artifacts are not as old, there is more information known about the quipus. Besides the 200 that were found, the Spanish conquistadors destroyed a lot of the quipus. Though the quipus were used to collect data, document census records, calendrical information, military organization, etc., the Spanish conquistadors thought of the quipus as something that represented the Inca religion. April Holloway claims in her article that, “the conquistadors were also attempting to convert the indigenous people to Roman Catholicism. The Inca religion was considered idolatry, anything that represented or was used by them was an attempt to disregard Catholic conversion.” Unlike the previous artifacts, a lot is known about the quipu and a lot of different samples were found showing the variation among them.

We learn and need math for many reasons, and societies develop systems to represent their idea of mathematics. The three artifacts that I found and learned about (including Plimpton 322), were completely different in how they were created and what they were created for. The oldest artifact that has been found is the Lebombo bone from 35,000B.C. It used a method of carving lines (the bone had 29 lines carved on it.) One of the oldest mathematical systems is still used today to teach children how to count. Thousands of years later in 1800 B.C., Plimpton 322 was created. Plimpton 322 was advanced compared to the Lebombo bone. It used a numeric system which expressed Pythagorean triples. The numeric system on the tablet was represented by symbols and used a base 60. Similar to the Lebombo bone, it had a system of math written on the artifacts. The most interesting and different artifact was the quipu which were created around 650 A.D. It is not similar to any math system that we use today. Every detail (like the material and color) about the quipu represented something significant. There are possibly many other mathematical artifacts in the world. How they were created and how they were used in their societies reflects how math was something civilizations have always needed. Artifacts, like the Lebombo bone, Plimpton 322, and the Quipu, reflect how they each could have shaped and develop the math that we used today.

Mathematicians might be extremely critical, obsessive, analytical, and somewhat strange people because of math. The mathematicians (I know of) in history books were either strange or had some very peculiar idiosyncrasies. Coming to touch a vast picture of the universe through mathematics may have changed who they were in a fundamental way that could not be undone, forever banishing them to strangeness.

Mathematicians’ uniqueness permeates history, so much that the average person is familiar with the reclusive/temperamental nature of Isaac Newton. I think that most of the world doesn’t know the story of Archimedes who died for a proof he was working on in the sand (Seife 52). This example of Archimedes was abnormal because most people wouldn’t be willing to have died for knowledge.

Some more lesser-known examples are Johannes Kepler and how he believed in the cult ideals of Pythagoras. Kepler believed in the sacredness of the ratios found in the world so much that it inhibited his mathematics in the area of orbital ellipses for a time (Maor 52). Hypatia believed that geometry had mythical meaning and was considered a cultist during her time (44 Derbyshire). She ended up being killed by an angry Christian mob for her abnormal beliefs (45 Derbyshire). Sophie Germain, being the awesome mathematician of all time, studied mathematics even when she was in pain and dying from breast cancer (Wikipedia). One last example was Mary Fairfax Somerville, who enjoyed mathematics so much that she continued to study even though her sister died and her parents forbade her from studying math (Wikipedia).

Our textbook contains many interesting mathematicians as well. Georg Cantor fathered set theory as the foundation for numbers and investigator of infinity, and had multiple psychological meltdowns during his life (75). Carl Friedrich Gauss was brutally honest and mentally crushed most of the people around him with his words. A simple example of this was János Bolyai (Laubenbacher and Pengelley 15). János Bolyai sent a message to Gauss telling him of his discovery of a non-Euclidean geometry and Gauss replied that he had already discovered this, which crushed Bolyai (Wikipedia). Gauss solved an extremely difficult summation problem when he was only ten years old, shocking the hell out of the people who knew him (Kaplan and Kaplan 31). Bernhard Bolzano defied the status quo to speak his mind to the point that he was banished from teaching or publishing his work in science and mathematics (Laubenbacher and Pengelley 70).

Pop culture doesn’t seem like a place where one would encounter mathematics, but movies have been made showcasing the quirky personalities of mathematicians. Probably the most popular in America are “A Beautiful Mind” and “Good Will Hunting.” John Nash is the mathematician that “A Beautiful Mind” was based on and he was a schizophrenic (Wikipedia).

I can tell when someone has seen the movie “A Beautiful Mind” by observing the other person writing on a window. It is very likely that they have done this to honor and respect the man or they just wanted others to perceive them as being smart. According to Nash in an interview in the documentary “John Nash: Documentary on the Beautiful and Insane Mind of John Nash,” he did this one time because he ran out of room on his whiteboard for an equation. A logical move to save paper/money by Nash has since then caused a lot of illogical, unnecessarily dirtied windows. Who would’ve known that logical thinking could lead to so much illogical behavior.… This also showed that what may seem strange to an outside viewer may be the completely logical move under the given circumstance for the party involved with the outcome.

The movie “N is a Number” is another movie based on a very unique mathematician, specifically Paul Erdos—a drug addict. Erdos used amphetamines as a strategy to improve his productivity in exploring mathematics, and his strategy worked (Wikipedia). There are many more movies with equally weird brilliant people with a propensity towards mathematics.

Not all mathematicians have been strange, but most have (opinion). It is also important to realize that there have been many mathematicians or pseudo mathematicians in history. So, the probability of having some peculiar folk is highly likely. When one has compared the history of other disciplines to mathematics—from my own biased sampling—one will find that there tends to be a greater reported quantity of strangeness amongst numberphiles. It is also important to point out that even though math minded people may act bizarre in some aspects, overall in the rote events of life, they tend to have made wiser and more logical choices than the average person (this is purely my opinion). We could also count our Professor Dr. Evelyn Lamb as another living example of a mathematician’s uniqueness.

This blog was not meant to propagate the idea that one must be born with an aptitude for mathematics. This blog also doesn’t intend to further the erroneous belief that one must have the right brain to be a mathematician, or that a hard-worker and dedication person couldn’t be a mathematician without some given aptitude (Klemm). The idea of this paper was that anyone who seeks the knowledge of mathematics successfully was changed mentally in such a way that most societies would deem abnormal.

On behalf of all the students to ever take a history of mathematics course thank you kooky wonky masters of the universe for making our textbook even more interesting than it would otherwise be.

P.S.

No formal scientific studies could be found on the common behaviors of mathematicians.

Works Cited

Derbyshire, John. The Unknown Quantity: A Brief History of Algebra. Washington, DC: Joseph

The angles of a triangle must add up to 180°. This is a simple fact that you were probably taught fairly early in your math career. It’s been known for millennia and is pretty simple to prove: for a right triangle, assume we have two parallel lines, one line perpendicular to them, and a fourth line between one of the intersections and an arbitrary non-intersection point on the opposite line as shown below.

This makes a triangle with one right angle, C, and two acute angles, A and B. We also need to consider angle D, the complementary angle to A. We know that A+D has to be 90° since they sum together to make a right angle, so the measure of angle D must be 90° – A. Since D and B are alternate interior angles with respect to the parallel lines and the red transverse line (remember all those awful congruence theorems you learned in your high school geometry class?) they have to be congruent angles. This means that the measure of angle B has to be 90°-A as well. So if we sum up the angles inside the triangle, A + B + C = A + 90° – A + 90° = 180° + A – A = 180°. The proofs for acute and obtuse triangles are similar, but a bit more complicated so we won’t go through them. The point is, we proved it! Triangles have to have 180°, right? Wrong.

The proof we used—and indeed all proofs that triangles must have 180° inside them—relies in some way on an infamous postulate used by Euclid around 300 BCE that says (more or less) that given a line and a point not on that line there is exactly one line through the point that does not intersect the original line. This postulate, though reasonable sounding, foiled mathematicians for thousands of years. Despite attempt after attempt to prove this postulate, no one was ever able to succeed. In fact, it was eventually proven that there IS no proof of this persnickety postulate. The angry mathematicians, having been foiled by this simple-yet-unprovable statement, began to consider what would happen if, indeed, it were not true. What would happen if, for example, there were an infinite number of lines through the point that didn’t intersect the original line? This line of questioning led to the discovery of hyperbolic geometry: a world where there are infinitely many parallels to a line through a given point off the line.

One of the many interesting aspects of hyperbolic geometry is that triangles don’t have to have 180°—In fact, they must have less than 180° (otherwise they could be a triangle in spherical or euclidean geometry). These triangles can still tessellate a plane though! In one particular representation of hyperbolic space, called a Poincaré disk, this tessellation would look like the image below.

The Poincaré disk is a way to show the hyperbolic plane on a circle. The idea is that straight lines are represented as curves from one side of the circle to another with the intention of preserving angles without necessarily preserving lengths. These curves must be circles that intersect the boundary of, or must be diameters of, the disk. The result is that each triangle in the picture above is the same size! From the large-looking central triangles to the itsy bitsy ones on the edge, each triangle would have exactly the same area in a hyperbolic space.

M.C. Escher was a Dutch artist whose graphics are widely known for their otherworldly bizarre mathematics. Stairs that led up to themselves and water that flowed in a ring are just two examples of his pieces, enacted with an almost formulaic mathematical exactness. He is well known in scientific communities for the diagramesque works of art.

You may be asking what this little Dutch artist has to do with our discussion of “curved” triangles. Well, Escher had become somewhat famous for using tessellations in his work. Creating shapes, especially in the shape of animals, which would tessellate all the way across the pieces, forming a lattice of cells that had only to be filled with a clever image. In the early 1950s, he became curious about finding different ways to “draw” infinity on a page. A letter from a friend came to him with some of these Poincaré tilings in the hyperbolic plane and became enamored with them. The images in the letter were a type of tiling denoted by {p,q} that was a tiling of p-gons with q of them meeting at each vertex. These images of hyperbolic tilings inspired Escher to create his Circle Limit series in 1959 and 1960. Circle Limit III was inspired in particular by the {8,3} tiling—4 octagons meeting at every vertex, and is a beautiful reimagining of the tiling with fish in place of the triangles.

Circle Limit III by M.C. Escher. His other work, including the other Circle Limits, can be found at http://www.mcescher.com.

Escher’s works seem to represent the very nature of the hyperbolic plane that we have talked about. After all, in a world where there are an infinite number of parallel lines, why couldn’t I draw infinite fishes on a page?